Trigonometry Examples

$\sin (\arctan (3 x))$

Step 1

Draw a triangle in the plane with vertices $(1, 3 x)$ , $(1, 0)$ , and the origin. Then $\arctan (3 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 3 x)$ . Therefore, $\sin (\arctan (3 x))$ is $\frac{3 x}{\sqrt{1 + {(3 x)}^{2}}}$ .

$\frac{3 x}{\sqrt{1 + {(3 x)}^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $3 x$ .

$\frac{3 x}{\sqrt{1 + 3^{2} x^{2}}}$

Step 2.2

Raise $3$ to the power of $2$ .

$\frac{3 x}{\sqrt{1 + 9 x^{2}}}$

Step 3

Multiply $\frac{3 x}{\sqrt{1 + 9 x^{2}}}$ by $\frac{\sqrt{1 + 9 x^{2}}}{\sqrt{1 + 9 x^{2}}}$ .

$\frac{3 x}{\sqrt{1 + 9 x^{2}}} \cdot \frac{\sqrt{1 + 9 x^{2}}}{\sqrt{1 + 9 x^{2}}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 x}{\sqrt{1 + 9 x^{2}}}$ by $\frac{\sqrt{1 + 9 x^{2}}}{\sqrt{1 + 9 x^{2}}}$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{\sqrt{1 + 9 x^{2}} \sqrt{1 + 9 x^{2}}}$

Step 4.2

Raise $\sqrt{1 + 9 x^{2}}$ to the power of $1$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{\sqrt{1 + 9 x^{2}}}^{1} \sqrt{1 + 9 x^{2}}}$

Step 4.3

Raise $\sqrt{1 + 9 x^{2}}$ to the power of $1$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{\sqrt{1 + 9 x^{2}}}^{1} {\sqrt{1 + 9 x^{2}}}^{1}}$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{\sqrt{1 + 9 x^{2}}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{\sqrt{1 + 9 x^{2}}}^{2}}$

Step 4.6

Rewrite ${\sqrt{1 + 9 x^{2}}}^{2}$ as $1 + 9 x^{2}$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 9 x^{2}}$ as ${(1 + 9 x^{2})}^{\frac{1}{2}}$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{({(1 + 9 x^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{(1 + 9 x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{(1 + 9 x^{2})}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{(1 + 9 x^{2})}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{3 x \sqrt{1 + 9 x^{2}}}{{(1 + 9 x^{2})}^{1}}$

Step 4.6.5

Simplify.

$\frac{3 x \sqrt{1 + 9 x^{2}}}{1 + 9 x^{2}}$